A treatise on the theory of Bessel functions, by G. N. Watson.

398

THEORY OF BESSEL FUNCTIONS

[CHAP. XIII

round a contour consisting of the real axis and a large semicircle above it, that
(dtA l ' f t ~ _ _ _ _ _ _ _ _ _ _ _ _ _ d t.
{1- cos (at)} f1 - cos (!t)} dt | sin (at) sin (a3t)
Hence the triple integral under consideration is equal to the triple integral evaluated in
proving (1), and consequently (2) is established in the same way as (1).
The reader will prove in like manner that, if R (/) and R (v) both exceed 2, then
(3) Ho (t) Jv (t) 2v - I
JO tL+J^-l /- 2y+V( + v-l)r(,+ ) r(v —)
and this may be extended over the range of values of e and v for which R (v) > 2 and
R(t+v)> 1.
H -1, (t) Hg ( t) Jr t)
The integrals f H (t (t) ) (t
Jo t+v+2 t#+v+i'
may be evaluated in a similar manner, but the results are of no great interest*.
13'4. The discontinuous integral of Weber and Schctfheitlin.
The integral
i" J, (at) J h (bt)
Jo t r
in which a and b are supposed to be positive to secure convergence at the
upper limit, was investigated by Weber, Journal fuir Math. LXXV. (1873),
pp. 75-80, in several special cases, namely,
(i) %=/ =0, IV= 1, (ii), = +-.
The integral was evaluated, for all values of X, /, and v for which it is convergent, by
Soninet, Math. Ann. xvi. (1880), pp. 51-52; but he did not examine the integral in very
great detail, nor did he lay any stress on the discontinuities which occur when a and b
become equal. Some years later the integral was investigated very thoroughly by Schafheitlin, but his preliminary analysis rests to a somewhat undue extent on the theory of
linear differential equations.
The special case in which X==0 was discussed in 1895 by Gubler~ who used a very
elegant transformation of contour integrals; unfortunately, however, it seems impossible
to adapt Gubler's analysis to the more general case in which X - 0. The analysis in the
special case will be given subsequently (~ 13'44).
* Some related integrals have been evaluated by Siemon, Programnn, Luisenschule, Berlin,
1890 [Jahrbuch iiber die Fortschritte der Math. 1890, p. 341].
+ See also ~ 13'43 in connexion with the researches of Gegenbauer, Wiener Sitzungsberichte,
LXXXVIII. (2), (1884), pp. 990-991.
+ Math. Ann. xxx. (1887), pp. 161 —178. The question of priority is discussed by Sonine,
Math. Ann. xxx. (1887), pp. 582-583, and by Schafheitlin, Math.. A xxx.. (1888), p. 156.
~ Math. Ann. XLVIII. (1897), pp. 37-48. See also Graf and Gubler, Einleitung in die Theorie
der Bessel'schen Funktionen, ii. (Bern, 1900), pp. 136-148.